Finding average rate of change of polynomials

- [Instructor] We are asked, What is the average rate of change of the function f, and this function is f up here, this is the definition of it, over the interval from negative two to three, and it's a closed interval because they put these brackets around it instead of parentheses, so that means it includes both of these boundaries. Pause this video and see if you can work through that. All right, now let's work through it together. So there's a couple of ways that we can conceptualize average rate of change of a function. One way to think about it is it's our change in the value of our function divided by our change in x, or it's our change in the value of our function per x on average. So you can view it as change in the value of the function, divided by your change in x. If you say that y is equal to f of x, you could also express it as change in y over change in x. On average, how much does the function change per unit change in x on average? And we could do this with a table, or we could try to conceptualize it visually. Let's just do this one with a table, and then we'll try to connect the dots a little bit with a visual. So if we have x here, and then if we have y is equal to f of x right over here, when x is equal to negative two, what is y going to be equal to, or what is f of negative two? Well, let's see. f of, so why is equal to f of negative two, which is going to be equal to negative eight, that's negative two to the third power, minus four times negative two, so that's minus negative eight, so that's plus eight. That equals zero. And then when x is equal to three, I'm going to the right end of that interval. Well, now y is equal to f of three, which is equal to 27, three to the third power, minus four times three, minus 12, which is equal to 15. So what is our change in y over our change in x over this interval? Well, our y went from zero to 15, so we have an increase of 15 in y. And what was our change in x? Well, we went from negative two to positive three, so we had a plus five change in x. So our change in x is plus five, and so our average rate of change with y with respect to x, or the rate of change of our function with respect to x, over the interval, is going to be equal to three. If you wanted to think about this visually, I could try to sketch this. So this is the x-axis, the y-axis, and our function does something like this. So at x equals negative two, f of x is zero, and then it goes up, and then it comes back down, and then it does something like this, it does something like this, and it does, and it was going before this, and so the interval that we care about, where we're going from negative two to three, which is right about there. So that's x equals negative two to x equals three, and so what we want to do, at the left end of the interval, our function is equal to zero, so we're at this point, right over there, I'll do this in a new color, at this point, right over there, and at the right end of our function, f of three is 15, so we are up here someplace. Let me connect the curve a little bit. We are going to be up there. And so when we're thinking about the average rate of change over this interval, we're really thinking about the slope of the line that connects these two points. So the line that connects these two points looks something like this. And we're just calculating what is our change in y, which is going to be this, our change in y, and we see that the value of our function increased by 15, divided by our change in x. So this right over here is our change in x, which we see went from negative two to three. That's going to be equal to five. So that's all we're doing when we're thinking about the average rate of change.